Headband not following rest of head
The problem is that although the head and head bands are assigned to the same vertexgroup they are not connected together, which results in giving them different "visual" effect when rotating the parent bone even though they have the same weight.

To fix this you will connect them. To do so, go to the Head Vertex Group, then select it. Then switch to Vertex Selection instead of Face selection. Then go to Mesh menu -> Vertices -> Connect Vertices. And voila it's done.

Steps:



Result:



Updated Blend file: